Stéphane Attal – författare

This is the ?rst in a series of three volumes dedicated to the lecture notes of the Summer School ”Open Quantum Systems” which took place at the Institut Fourier in Grenoble from June 16th to July 4th 2003. The contributions presented in these volumes are revised and expanded versions of the notes provided to the students during the School. Closed vs. Open Systems By de?nition, the time evolution of a closed physical systemS is deterministic. It is usually described by a differential equation x ? = X(x ) on the manifold M of t t possible con?gurations of the system. If the initial con?guration x ? M is known 0 then the solution of the corresponding initial value problem yields the con?guration x at any future time t. This applies to classical as well as to quantum systems. In the t classical case M is the phase space of the system and x describes the positions and t velocities of the various components (or degrees of freedom) ofS at time t. Inthe quantum case, according to the orthodoxinterpretation of quantum mechanics, M is a Hilbert space and x a unit vector – the wave function – describing the quantum t state of the system at time t. In both cases the knowledge of the state x allows t to predict the result of any measurement made onS at time t.

This volume is the second in a series of three volumes dedicated to the lecture notes of the summer school “Open Quantum Systems” which took place in the Institut Fourier in Grenoble, from June 16th to July 4th 2003. The contributions presented in thesevolumesarerevisedandexpandedversionsofthenotesprovidedtothestudents during the school. After the rst volume, developing the Hamiltonian approach of open quantum systems, this second volume is dedicated to the Markovian approach. The third volume presents both approaches, but at the recent research level. Open quantum systems A quantum open system is a quantum system which is interacting with another one. This is a general de nition, but in general, it is understood that one of the s- tems is rather “small” or “simple” compared to the other one which is supposed to be huge, to be the environment, a gas of particles, a beam of photons, a heat bath ... The aim of quantum open system theory is to study the behaviour of this coupled system and in particular the dissipation of the small system in favour of the large one.

This volume is the third and last of a series devoted to the lecture notes of the Grenoble Summer School on “Open Quantum Systems” which took place at the th th Institut Fourier from June 16 to July 4 2003. The contributions presented in this volumecorrespondtoexpanded versionsofthelecturenotesprovidedbytheauthors to the students of the Summer School. The corresponding lectures were scheduled in the last part of the School devoted to recent developments in the study of Open Quantum Systems. Whereas the rst two volumes were dedicated to a detailed exposition of the mathematical techniques and physical concepts relevant in the study of Open S- tems with noapriori pre-requisites, the contributions presented in this volume request from the reader some familiarity with these aspects. Indeed, the material presented here aims at leading the reader already acquainted with the basics in ? quantum statistical mechanics, spectral theory of linear operators,C -dynamical systems, and quantum stochastic differential equations to the front of the current research done on various aspects of Open Quantum Systems. Nevertheless, pe- gogical efforts have been made by the various authors of these notes so that this volume should be essentially self-contained for a reader with minimal previous - posure to the themes listed above. In any case, the reader in need of complements can always turn to these rst two volumes. The topics covered in these lectures notes start with an introduction to n- equilibrium quantum statistical mechanics.

This is the ?rst in a series of three volumes dedicated to the lecture notes of the Summer School ”Open Quantum Systems” which took place at the Institut Fourier in Grenoble from June 16th to July 4th 2003. The contributions presented in these volumes are revised and expanded versions of the notes provided to the students during the School. Closed vs. Open Systems By de?nition, the time evolution of a closed physical systemS is deterministic. It is usually described by a differential equation x ? = X(x ) on the manifold M of t t possible con?gurations of the system. If the initial con?guration x ? M is known 0 then the solution of the corresponding initial value problem yields the con?guration x at any future time t. This applies to classical as well as to quantum systems. In the t classical case M is the phase space of the system and x describes the positions and t velocities of the various components (or degrees of freedom) ofS at time t. Inthe quantum case, according to the orthodoxinterpretation of quantum mechanics, M is a Hilbert space and x a unit vector – the wave function – describing the quantum t state of the system at time t. In both cases the knowledge of the state x allows t to predict the result of any measurement made onS at time t.

This volume is the second in a series of three volumes dedicated to the lecture notes of the summer school “Open Quantum Systems” which took place in the Institut Fourier in Grenoble, from June 16th to July 4th 2003. The contributions presented in thesevolumesarerevisedandexpandedversionsofthenotesprovidedtothestudents during the school. After the rst volume, developing the Hamiltonian approach of open quantum systems, this second volume is dedicated to the Markovian approach. The third volume presents both approaches, but at the recent research level. Open quantum systems A quantum open system is a quantum system which is interacting with another one. This is a general de nition, but in general, it is understood that one of the s- tems is rather “small” or “simple” compared to the other one which is supposed to be huge, to be the environment, a gas of particles, a beam of photons, a heat bath ... The aim of quantum open system theory is to study the behaviour of this coupled system and in particular the dissipation of the small system in favour of the large one.